When we want to add or subtract two or more radical terms, first we have to verify whether they are having same number inside the radical sign or not.

The terms which are having same number inside the radical sign is known as like radicals.

Some times we will have a large number inside the radical sign, that time we have to split the number inside the radical sign as much as possible and we can take one common number for every two same numbers multiplying inside the radical sign and make it as like radicals.

Let us see some example problems to understand the concept.

Example 1 :

Simplify the radical expression given below

√3 + √12

Solution :

=  3 √3

Example 2 :

Simplify the radical expression given below

√75 + √25

Solution :

=  √75 + √25

Here the given two radical terms are not like terms.

=  √(5 x 5 x 5) + √(5 x 5)

=  5 √5 + 5

=  6 √5

Example 3 :

Simplify the radical expression given below

√7 + √49

Solution :

=  √7 + √49

Since 7 is prime number, we cannot split this hereafter. We can split 49.

=  √7 + √(7 x 7)

=  √7 + 7

Example 4 :

Simplify the radical expression given below

√5 + 2√5 - 5√5

Solution :

=  √5 + 2√5 - 5√5

Since √5 is common for all three terms, we are going to take √5 commonly from all the terms and simplify the numbers.

=  (1 + 2 - 5) √5

=  (3 - 5) √5

=  -2 √5

Example 5 :

Simplify the radical expression given below

√5 + 3 √7 - 4 √5 - 5 √7

Solution :

=  √5 + 3 √7 - 4 √5 - 5 √7

Now we have to group the like radicals.

=  √5 - 4 √5 + 3 √7  - 5 √7

=  (1 - 4) √5 + (3 - 5) √7

=  - 3 √5 - 2 √7

Example 6 :

Simplify the radical expression given below

3√3 + 4 √3 - √2

Solution :

=  3√3 + 4 √3 - √2

The first two terms are having √3. Hence they are like radicals, we can combine them.

=  (3 + 4) √3 - √2

=  7 √3 - √2

Example 7 :

Simplify the radical expression given below

2(√5 - √3) + 3(√3 - √5)

Solution :

=  2(√5 - √3) + 3(√3 - √5)

Distribute 2 for (√5 - √3) and distribute 3 for (√3 - √5).

=  2√5 - 2√3 + 3√3 - 3√5

Now we have to combine the like terms

=  2√5 - 3√5 - 2√3 + 3√3

=  (2 - 3)√5 + (-2 + 3)√3

=  -1√5 + 1√3

=  -√5 + √3

Example 8 :

Simplify the radical expression given below

√8 + √18

Solution :

=  √8 + √18

We can split 8 and 18 as much as possible to get factors.

=  √(2 x 2 x 2) + √(2 x 3 x 3)

=  2√2 + 3√2

=  (2 + 3) √2

=  5 √2

Example 9 :

Simplify the radical expression given below

√12w + √27w

Solution :

=  √12w + √27w

We can split 12 and 27 as much as possible to get factors.

=  √(2 x 2 x 3 x w) + √(3 x 3 x 3 x w)

=  2√(3 x w) + 3√(3 x w)

=  2√3w + 3√3w

=  (2 + 3)√3w

=  5√3w

Example 10 :

Simplify the radical expression given below

√45t^2 + √25t^3

Solution :

=  √45t^3 + √20t^3

We can split 45 and 25 as much as possible to get factors.

=  √(3 x 3 x 5 x t^3) + √(2 x 2 x 5 x t^3)

=  3 t √5t + 2 t√5t

=  (3 t + 2t) √5t

=  5t √5t

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